1. Field of the Invention
The present invention generally relates to discrete multi-tone communication systems, and more particularly to a discrete multi-tone communication system using discrete Hartley transform for modulation and demodulation.
2. The Prior Arts
Discrete multi-tone (DMT) is a multi-carrier modulation (MCM) technique commonly applied in wireline communications such as digital subscriber loops (xDSL) including ADSL, ADSL2, ADSL2+, and VDSL, and power-line communications such as HomePlug. The basic idea of DMT is that a large number of sinusoids (i.e., subcarriers) are modulated by complex-valued quadrature amplitude modulation (QAM) symbols derived from an input bit stream, and transmitted in parallel. Performing a modulation on the complex valued constellation points generates samples of the continuous-time signal to be transmitted during a DMT symbol period. At the receiver, the QAM symbols are recovered by performing a demodulation on the analog-to-digital-converted received signal. A typical DMT system for xDSL is shown in FIG. 1. As illustrated, discrete Fourier transform (DFT) and IDFT (i.e., Inversed DFT) are adopted for baseband demodulation and modulation for the DMT system in the receiving end (Rx) and transmitting end (Tx) (separated by the dashed line in FIG. 1) respectively. A pair of digital-to-analog (D/A) converter and an analog-to-digital (A/D) converter is also provided around the analog communication channel. A time-domain equalizer (TEQ) is used at the receiving end to shorten the channel dispersion and, thereby, minimize intersymbol interference (ISI) caused by channel distortion resulted from, for example, long loop length, gauge variation, and bridge-tap. In addition, to compensate the phase rotation and amplitude distortion existing on each subchannel in the frequency domain, a frequency-domain equalizer (FEQ) is deployed. Also, to guard against the ISI, cyclic prefix (CP) are added and removed at the transmitting end and the receiving end, respectively.
The modulation and demodulation operations performed by the IDFT and DFT can be expressed mathematically as follows:
                                          IDFT            ⁢                          :                        ⁢                                                  ⁢                          x              n                                =                                    1              N                        ⁢                                          ∑                                  k                  =                  0                                                  N                  -                  1                                            ⁢                                                X                  k                                ·                                  W                  N                                      -                    kn                                                                                      ,                  n          =          0                ,        1        ,        …        ⁢                                  ,                  N          -          1                                    (        1        )                                                      DFT            ⁢                          :                        ⁢                                                  ⁢                          Y              k                                =                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                          y                n                            ·                              W                N                kn                                                    ,                  k          =          0                ,        1        ,        …        ⁢                                  ,                  N          -          1                                    (        2        )            where
            W      N        =          ⅇ                        -          j                ⁢                              2            ⁢            π                    N                      ,Xk or Yk is the complex transmission symbol on the k-th subchannel in frequency domain, xn or yn is the n-th transmission sample in time domain, and N is the number of points for the IDFT/DFT.
The FEQ or, as it is used behind the DFT, the DFT-based FEQ can be implemented using various algorithms. For example, the MMSE (minimum mean-square error) algorithm tries to minimize the estimated error between the equalized signal and the transmitted signal. Among these algorithms, the LMS (least mean square) algorithm is most popular in terms of VLSI implementation. The LMS algorithm in the DFT-based FEQ consists of three operations: filtering, error estimation, and coefficient updating, expressed mathematically as follows:Filtering: Ŷk=Yk·Wk*=(Yk,R+jYk,I)·(Wk,R−jWk,I)  (3)Error estimation: Ek=Xk−Ŷk  (4)Coefficient Updating: Wk(n+1)=Wk(n)+μkYkEk*  (5)where the subscripts R and I represent the real part and imaginary part of a complex number respectively, the superscript asterisk (*) denotes the complex conjugate operation, Yk is the DFT output, Xk is the training symbol of the k-th subchannel of the DMT system, Wk is the FEQ coefficient of the k-th subchannel, and μk is the updating step-size of the FEQ for the k-th subchannel. An embodiment of the DFT-based FEQ is illustrated in FIG. 2.
Prior arts have suggested replacing the DFT with discrete Hartley transform (DHT) so as to reduce the computing complexity from complex to real multiplication involved in DFT, as DHT itself is real-valued operation. With such a substitution, the DFT-based DMT system shown in FIG. 1 would become a DMT system illustrated in FIG. 3, where the modulation at the transmitting end is realized by the inverse DHT (IDHT) with a preceding complex-to-real transformation (C2RT), and the demodulation at the receiving end is realized by DHT with a succeeding real-to-complex transformation (R2CT). C2RT is required to transform the complex symbol Xk into the real symbol Hk; and R2CT is required to transform the real symbol {tilde over (H)}k to the complex symbol Yk. Please note that the DFT-based FEQ still has to be adopted in the DHT-based DMT system of FIG. 3, as basically the FEQ architecture remains the same.
The modulation and demodulation operations performed by the IDHT and DHT can be expressed mathematically as follows:
                                          IDHT            ⁢                          :                        ⁢                                                  ⁢                          x              N                                =                                    1              N                        ⁢                                          ∑                                  k                  =                  0                                                  N                  -                  1                                            ⁢                                                H                  k                                ·                                  cas                  ⁡                                      (                                          2                      ⁢                      π                      ⁢                                                                                          ⁢                      n                      ⁢                                                                                          ⁢                                              k                        /                        N                                                              )                                                                                      ,                  n          =          0                ,        1        ,        …        ⁢                                  ,                  N          -          1                                    (        6        )                                                      DHT            ⁢                          :                        ⁢                                                  ⁢                                          H                ~                            k                                =                                    ∑                              n                =                0                                            N                -                1                                      ⁢                                          y                n                            ·                              cas                ⁡                                  (                                      2                    ⁢                    π                    ⁢                                                                                  ⁢                    n                    ⁢                                                                                  ⁢                                          k                      /                      N                                                        )                                                                    ,                  k          =          0                ,        1        ,        …        ⁢                                  ,                  N          -          1                                    (        7        )            where cas(·)=cos(·)+sin(·), and N is the number of points for the IDHT/DHT and is the same as that for the IDFT/DFT. Similarly, the C2RT and R2CT can be expressed as follows:C2RT: Hk=Xk,R−Xk,I  (8)R2CT: Yk=Yk,R+jYk,I={tilde over (H)}k,E−j{tilde over (H)}k,O  (9)where {tilde over (H)}k,E and {tilde over (H)}k,O are the even and odd parts of the {tilde over (H)}k respectively, which can be obtained by:
                                          H            ~                                k            ,            E                          =                                                            H                ~                            k                        +                                          H                ~                                            N                -                k                                              2                                    (        10        )                                                      H            ~                                k            ,            O                          =                                                            H                ~                            k                        -                                          H                ~                                            N                -                k                                              2                                    (        11        )            